Radial Motion of Two Mutually Attracting Particles

Abstract

A pair of masses or opposite-sign charges released from rest will move directly toward each other under the action of the inverse-distance-squared force of attraction between them. An exact expression for the separation distance as a function of time can only be found by numerically inverting the solution of a differential equation. A simpler, approximate formula can be obtained by combining dimensional analysis, Kepler's third law, and the familiar quadratic dependence of distance on time for a mass falling near Earth's surface. These exact and approximate results are applied to several interesting examples: the flight time and maximum altitude attained by an object fired straight upward from Earth's surface; the time required for an asteroid of known starting position and speed to cross Earth's orbit if it is bearing toward the Sun; and the collision time of two oppositely charged particles starting from rest.

Satellite splat: an inelastic collision with a surface-launched projectile

“…Your first inclination may be to fire a projectile vertically (i.e., radially) from the planet's surface. The projectile's path is thus a degenerate orbit [4] with periapsis distance r peri = 0 and specific angular momentum h P = 0. There are two questions we shall answer for this case:…”

Satellite splat II: an inelastic collision with a surface-launched projectile and the maximum orbital radius for planetary impact

“…As long as ε < 0, which requires k < 5 from equation (3), it will subsequently fall into the planet with an impact directly below the collision point. Such bound radial trajectories are discussed in [4].…”